Square root of 3

The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is more precisely called the principal square root of 3, to distinguish it from the negative number with the same property. It is denoted by √3.

1.73205080756887729352744634150587236694280525381038062805580… (sequence A002194 in the OEIS)

As of December 2013, its numerical value in decimal has been computed to at least ten billion digits.[1]

The fraction 97/56 (7000173214285700000♠1.732142857…) for the square root of three can be used as an approximation. Despite having a denominator of only 56, it differs from the correct value by less than 1/10,000 (approximately 6995920000000000000♠9.2×10−5). The rounded value of 1.732 is correct to within 0.01% of the actual value.

The square of √3 can be replaced by 3. As m/n is multiplied by n, their product equals m:

3n−mqm−nq{\displaystyle {\frac {3n-mq}{m-nq}}}

Then √3 can be expressed in lower terms than m/n (since the first step reduced the sizes of both the numerator and the denominator, and subsequent steps did not change them) as 3n − mq/m − nq, which is a contradiction to the hypothesis that m/n was in lowest terms.[3]

Since the left side is divisible by 3, so is the right side, requiring that m be divisible by 3. Then, m can be expressed as 3k:

3n2=(3k)2=9k2{\displaystyle 3n^{2}=(3k)^{2}=9k^{2}}

Therefore, dividing both terms by 3 gives:

n2=3k2{\displaystyle n^{2}=3k^{2}}

Since the right side is divisible by 3, so is the left side and hence so is n. Thus, as both n and m are divisible by 3, they have a common factor and m/n is not a fully reduced fraction, contradicting the original premise.

The square root of 3 is equal to the length between parallel sides of a regular hexagon with sides of length 1.

The square root of 3 can be found as the leg length of an equilateral triangle that encompasses a circle with a diameter of 1.

If an equilateral triangle with sides of length 1 is cut into two equal halves, by bisecting an internal angle across to make a right angle with one side, the right angle triangle's hypotenuse is length one and the sides are of length 1/2 and √3/2. From this the trigonometric function tangent of 60° equals √3, and the sine of 60° and the cosine of 30° both equal √3/2.

In power engineering, the voltage between two phases in a three-phase system equals √3 times the line to neutral voltage. This is because any two phases are 120° apart, and two points on a circle 120 degrees apart are separated by √3 times the radius (see geometry examples above).